Why isn't Vuze running when I restart my computer?

Vuze is an application and requires that you start it in order for it to run. Currently it doesn't have an option to run as a service (on Windows), which could start automatically when the system is restarted. So when you shut down your computer, the application will quit, and when you restart your computer you will need to launch the application when you want it to be running. You can do this from the application dock (Mac) or the Start menu (Windows).

If you want the application to launch every time you reboot your machine, you can set this by using control-click on the dock icon (Mac), or create a shortcut to your azureus.exe executable in the Startup folder under All Applications in your Start menu (Windows)